During inelastic collision between two bodies, which of the following quantities always remain conserved?
A. Total kinetic energy
B. Total mechanical energy
C. Total inner momentum
D. Speed of each body

Hint: Characteristics of inelastic collision that is whenever inelastic collision there is loss in kinetic energy due to which speed change.

Complete step by step answer:
Collision: When two bodies either physically collide against each other or if the path of one is affected by force of another, the collision is said to occur.
For collision to take place, actual physical contact is not a necessity. For example is Rutherford $\alpha $-scattering experiment, $\alpha $particle is scattered due to repulsion between $\alpha $-particles and nucleus from a distance and here we say $\alpha $particles have suffered collision.
Elastic collision: If there is no less of kinetic energy during the collision it is said to be elastic
For example-collision between substance particles.
Inelastic collision- When there is a loss of kinetic energy during collision, it is said to be inelastic.
For example collision between two vehicles
Characteristics of inelastic collision:
Total energy of the system is conserved.
The kinetic energy is not conserved as there is loss of kinetic energy.
There is conservation of momentum
Non-conservation forces are involved.
Some part of mechanical energy is converted into light, heat etc.
So from the above discussion, it is clear that during inelastic collision between two bodies, total mechanical energy and total linear momentum is conserved.

So, the correct answers are “Option B and C”.

Additional Information:
Characteristics of elastic collision:
Total linear momentum is conserved.
Total energy is conserved.
Kinetic energy is also conserved.
Conservative forces are involved
No conversion of mechanical energy into light, heat, sound etc.

Note:
Since, no external forces are acting there on colliding bodies use ${F_{ext}} = 0$ and $F = \dfrac{{dp}}{{dt}}$
$ \Rightarrow p = cons\tan t$
Hence, total linear momentum is conserved.